Properties

Label 2-2013-1.1-c3-0-262
Degree $2$
Conductor $2013$
Sign $-1$
Analytic cond. $118.770$
Root an. cond. $10.8982$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.38·2-s + 3·3-s − 6.07·4-s + 18.9·5-s + 4.16·6-s − 17.1·7-s − 19.5·8-s + 9·9-s + 26.2·10-s + 11·11-s − 18.2·12-s − 13.0·13-s − 23.7·14-s + 56.7·15-s + 21.5·16-s − 14.5·17-s + 12.4·18-s − 85.6·19-s − 114.·20-s − 51.4·21-s + 15.2·22-s + 89.8·23-s − 58.5·24-s + 232.·25-s − 18.0·26-s + 27·27-s + 104.·28-s + ⋯
L(s)  = 1  + 0.490·2-s + 0.577·3-s − 0.759·4-s + 1.69·5-s + 0.283·6-s − 0.925·7-s − 0.862·8-s + 0.333·9-s + 0.829·10-s + 0.301·11-s − 0.438·12-s − 0.277·13-s − 0.453·14-s + 0.976·15-s + 0.336·16-s − 0.206·17-s + 0.163·18-s − 1.03·19-s − 1.28·20-s − 0.534·21-s + 0.147·22-s + 0.814·23-s − 0.498·24-s + 1.86·25-s − 0.136·26-s + 0.192·27-s + 0.703·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $-1$
Analytic conductor: \(118.770\)
Root analytic conductor: \(10.8982\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2013,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
11 \( 1 - 11T \)
61 \( 1 - 61T \)
good2 \( 1 - 1.38T + 8T^{2} \)
5 \( 1 - 18.9T + 125T^{2} \)
7 \( 1 + 17.1T + 343T^{2} \)
13 \( 1 + 13.0T + 2.19e3T^{2} \)
17 \( 1 + 14.5T + 4.91e3T^{2} \)
19 \( 1 + 85.6T + 6.85e3T^{2} \)
23 \( 1 - 89.8T + 1.21e4T^{2} \)
29 \( 1 + 302.T + 2.43e4T^{2} \)
31 \( 1 + 58.8T + 2.97e4T^{2} \)
37 \( 1 - 183.T + 5.06e4T^{2} \)
41 \( 1 - 289.T + 6.89e4T^{2} \)
43 \( 1 + 74.0T + 7.95e4T^{2} \)
47 \( 1 + 271.T + 1.03e5T^{2} \)
53 \( 1 + 683.T + 1.48e5T^{2} \)
59 \( 1 - 70.5T + 2.05e5T^{2} \)
67 \( 1 + 394.T + 3.00e5T^{2} \)
71 \( 1 + 280.T + 3.57e5T^{2} \)
73 \( 1 + 661.T + 3.89e5T^{2} \)
79 \( 1 + 1.18e3T + 4.93e5T^{2} \)
83 \( 1 - 406.T + 5.71e5T^{2} \)
89 \( 1 - 1.41e3T + 7.04e5T^{2} \)
97 \( 1 + 709.T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.830833851077350030033547647491, −7.60136630716362242894106022653, −6.48741647085795970877931859005, −6.05114508934412502257095518759, −5.20362453293424494569821854826, −4.31152018369694011843798288235, −3.32205083351727649947705678649, −2.52497405622696116485699739201, −1.50599346609920076618076711508, 0, 1.50599346609920076618076711508, 2.52497405622696116485699739201, 3.32205083351727649947705678649, 4.31152018369694011843798288235, 5.20362453293424494569821854826, 6.05114508934412502257095518759, 6.48741647085795970877931859005, 7.60136630716362242894106022653, 8.830833851077350030033547647491

Graph of the $Z$-function along the critical line